Tutorial ExerciseUse the given transformation to evaluate the given integral, where Ris the triangular region with vertices (0,0),(2,1), and (1,2).L=R(x-3y)dA;,x=2u+v,y=u+2vPart 1of6For the transformation x=2u+v,y=u+2v, the Jacobian isdel(x,y)del(u,v)=|[delxdelu,delxdelv],[delydelu,delydelv]|=|[2,1],[1,2,2]|=3,{:3Also,x-3y=(2u+v)-3(u+2v)=-u-5Part 2of6To find the region Sin the uv-plane which corresponds toR,we find the corresponding boundaries. The line though (0,0) and (2,1)isy=12,xx and this is the image ofv=Part 3of6The line through (2,1) and (1,2)isy=, and this is the image ofv=The line through (0,0) and (1,2)isy=, and this is the image ofu=0.Part 4of6Now we can say thatR(x-3y)dA=0101-u(-u-5v)|3|dvduPart 5of6We have